Sharp reverse fractional Hausdorff inequality on power-weighted Lebesgue spaces

Our main focus in this paper is to explore the mapping properties for the n-dimensional fractional Hausdorff operator \(H_{\Phi ,\beta }\) $H_{\mathrm{\Phi },\beta }$ from \(L^{p}(\mathbb {R}^{n},|x|^{\alpha })\) $L^p({\mathbb{R}}^n{,|x|}^{\alpha })$ to \(L^{q}(\mathbb {R}^{n},|x|^{\gamma })\) $L^q({\mathbb{R}}^n{,|x|}^{\gamma })$ , where \(p,q<1~(p,q\ne 0)\) $p,q<1(p,q\ne 0)$ , \(\alpha ,\gamma \in \mathbb {R}\) $\alpha ,\gamma \in \mathbb{R}$ , \(0\le \beta <n\) $0\le \beta <n$ and \(\Phi \) $\mathrm{\Phi }$ is a nonnegative measurable function on \(\mathbb {R}^n\) ${\mathbb{R}}^n$ . For \(p,q<1~(p,q\ne 0)\) $p,q<1(p,q\ne 0)$ satisfying some additional assumptions, we give sufficient conditions for the validity of the reverse fractional Hausdorff inequality \(\left\| H_{\Phi ,\beta }f\right\| _{L^{q}(\mathbb {R}^{n},|x|^{\gamma })}\ge C\Vert f\Vert _{L^{p}(\mathbb {R}^{n},|x|^{\alpha })}\) ${∥H_{\mathrm{\Phi },\beta },f∥}_{L^q({\mathbb{R}}^n{,|x|}^{\gamma })}\ge C{\Vert f\Vert }_{L^p({\mathbb{R}}^n{,|x|}^{\alpha })}$ for some positive constant C and all nonnegative functions \(f\in L^{p}(\mathbb {R}^{n},|x|^{\alpha })\) $f\in L^p({\mathbb{R}}^n{,|x|}^{\alpha })$ . For the particular case \(0<p=q<1\) $0<p=q<1$ , we obtain the sharp reverse fractional Hausdorff inequality. As applications, we establish the sharp reverse inequalities for the n-dimensional fractional Hardy operator and its adjoint operator, and also the n-dimensional fractional Hardy–Littlewood–Pólya operator on power-weighted Lebesgue spaces.